| Description |
1 online resource |
| Note |
Includes index. |
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Print version record. |
| Contents |
Front cover -- Half title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgments -- Chapter 1 Introduction -- 1.1 Modeling and simulation -- 1.1.1 Boundary and initial value problems -- 1.1.2 Boundary value problems -- 1.2 Solution methods -- Chapter 2 Mathematical modeling of physical systems -- 2.1 Introduction -- 2.2 Governing equations of structural mechanics -- 2.2.1 External forces, internal forces, and stress -- 2.2.2 Stress transformations -- 2.2.3 Deformation and strain -- 2.2.4 Strain compatibility conditions -- 2.2.5 Generalized Hooke's law -- 2.2.6 Two-dimensional problems -- 2.2.7 Balance laws -- 2.2.8 Boundary conditions -- 2.2.9 Total potential energy of conservative systems -- 2.3 Mechanics of a flexible beam -- 2.3.1 Equation of motion of a beam -- 2.3.2 Kinematics of the Euler-Bernoulli beam -- 2.3.3 Stresses in an Euler-Bernoulli beam -- 2.3.4 Kinematics of the Timoshenko beam -- 2.3.5 Stresses in a Timoshenko beam -- 2.3.6 Governing equations of the Euler-Bernoulli beam theory -- 2.3.7 Governing equations of the Timoshenko beam theory -- 2.4 Heat transfer -- 2.4.1 Conduction heat transfer -- 2.4.2 Convection heat transfer -- 2.4.3 Radiation heat transfer -- 2.4.4 Heat transfer equation in a one-dimensional solid -- 2.4.5 Heat transfer in a three-dimensional solid -- 2.5 Problems -- References -- Chapter 3 Integral formulations and variational methods -- 3.1 Introduction -- 3.2 Mathematical background -- 3.2.1 Divergence theorem -- 3.2.2 Green-Gauss theorem -- 3.2.3 Integration by parts -- 3.2.4 Fundamental lemma of calculus of variations -- 3.2.5 Adjoint and self-adjoint operators -- 3.3 Calculus of variations -- 3.3.1 Variation of a functional -- 3.3.2 Functional derivative -- 3.3.3 Properties of functionals -- 3.3.4 Properties of the variational derivative. |
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3.3.5 Euler-Lagrange equations and boundary conditions -- 3.4 Weighted residual integral and the weak form of boundary value problems -- 3.4.1 Weighted residual integral -- 3.4.2 Boundary conditions -- 3.4.3 The weak form -- 3.4.4 Relationship between the weak form and functionals -- 3.5 Method of weighted residuals -- 3.5.1 Rayleigh-Ritz method -- 3.5.2 Galerkin method -- 3.5.3 Polynomials as basis functions for Rayleigh-Ritz and Galerkin methods -- 3.6 Problems -- References -- Chapter 4 Finite element formulation of one-dimensional boundary value problems -- 4.1 Introduction -- 4.1.1 Boundary value problem -- 4.1.2 Spatial Discretization -- 4.2 A second order, nonconstant coefficient ordinary differential equation over an element -- 4.2.1 Deflection of a one-dimensional bar -- 4.2.2 Heat transfer in a one-dimensional domain -- 4.3 One-dimensional interpolation for finite element method and shape functions -- 4.3.1 C0 continuous, linear shape functions -- 4.3.2 C0 continuous, quadratic shape functions -- 4.3.3 General form of C0 shape functions -- 4.3.4 One-dimensional, Lagrange interpolation functions -- 4.4 Equilibrium equations in finite element form -- 4.4.1 Element stiffness matrix for constant problem parameters -- 4.4.2 Element stiffness matrix for linearly varying problem parameters a, p, and q -- 4.5 Recovering specific physics from the general finite element form -- 4.6 Element assembly -- 4.7 Boundary conditions -- 4.7.1 Natural boundary conditions -- 4.7.2 Essential boundary conditions -- 4.8 Computer implementation -- 4.8.1 Main-code -- 4.8.2 Element connectivity table -- 4.8.3 Element assembly -- 4.8.4 Boundary conditions -- 4.9 Example problem -- 4.10 Problems -- Chapter 5 Finite element analysis of planar bars and trusses -- 5.1 Introduction -- 5.2 Element equilibrium equation for a planar bar -- 5.2.1 Problem definition. |
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5.2.2 Weak form of the boundary value problem -- 5.2.3 Total potential energy of the system -- 5.2.4 Finite element form of the equilibrium equations of an elastic bar -- 5.3 Finite element equations for torsion of a bar -- 5.4 Coordinate transformations -- 5.4.1 Transformation of unit vectors between orthogonal coordinate systems -- 5.4.2 Transformation of equilibrium equations for the one-dimensional bar element -- 5.5 Assembly of elements -- 5.6 Boundary conditions -- 5.6.1 Formal definition -- 5.6.2 Direct assembly of the active degrees of freedom -- 5.6.3 Numerical implementation of the boundary conditions -- 5.7 Effects of initial stress or initial strain -- 5.7.1 Thermal stresses -- 5.7.2 Initial stresses -- 5.8 Postprocessing: Computation of stresses and reaction forces -- 5.8.1 Computation of stresses in members -- 5.8.2 Reaction forces -- 5.9 Error and convergence in finite element analysis -- Problems -- Reference -- Chapter 6 Euler-Bernoulli beam element -- 6.1 Introduction -- 6.2 C1-Continuous interpolation function -- 6.3 Element equilibrium equation -- 6.3.1 Problem definition -- 6.3.2 Weak form of the boundary value problem -- 6.3.3 Total potential energy of a beam element -- 6.3.4 Finite element form of the equilibrium equations of an Euler-Bernoulli beam -- 6.4 General beam element with membrane and bending capabilities -- 6.5 Coordinate transformations -- 6.5.1 Vector transformation between orthogonal coordinate systems in a two-dimensional plane -- 6.5.2 Transformation of equilibrium equations for the Euler-Bernoulli beam element with axial deformation -- 6.6 Assembly, boundary conditions, and reaction forces -- 6.7 Postprocessing and computation of stresses in members -- Example 6.1 -- Problems -- Reference -- Chapter 7 Isoparametric elements for two-dimensional elastic solids -- 7.1 Introduction. |
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7.2 Solution domain and its boundary -- 7.2.1 Outward unit normal and tangent vectors along the boundary -- 7.3 Equations of equilibrium for two-dimensional elastic solids -- 7.4 General finite element form of equilibrium equations for a two-dimensional element -- 7.4.1 Variational form of the equation of equilibrium -- 7.4.2 Finite element form of the equation of equilibrium -- 7.5 Interpolation across a two-dimensional domain -- 7.5.1 Two-dimensional polynomials -- 7.5.2 Two-dimensional shape functions -- 7.6 Mapping between general quadrilateral and rectangular domains -- 7.6.1 Jacobian matrix and Jacobian determinant -- 7.6.2 Differential area in curvilinear coordinates -- 7.7 Mapped isoparametric elements -- 7.7.1 Strain-displacement operator matrix, [B] -- 7.7.2 Finite element form of the element equilibrium equations for a Q4-element -- 7.8 Numerical integration using Gauss quadrature -- 7.8.1 Coordinate transformation -- 7.8.2 Derivation of second-order Gauss quadrature -- 7.8.3 Integration of two-dimensional functions by Gauss quadrature -- 7.9 Numerical evaluation of the element equilibrium equations -- 7.10 Global equilibrium equations and boundary conditions -- 7.10.1 Assembly of global equilibrium equation -- 7.10.2 General treatment of the boundary conditions -- 7.10.3 Numerical implementation of the boundary conditions -- 7.11 Postprocessing of the solution -- References -- Chapter 8 Rectangular and triangular elements for two-dimensional elastic solids -- 8.1 Introduction -- 8.1.1 Total potential energy of an element for a two-dimensional elasticity problem -- 8.1.2 High-level derivation of the element equilibrium equations -- 8.2 Two-dimensional interpolation functions -- 8.2.1 Interpolation and shape functions in plane quadrilateral elements -- 8.2.2 Interpolation and shape functions in plane triangular elements. |
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8.3 Bilinear rectangular element (Q4) -- 8.3.1 Element stiffness matrix -- 8.3.2 Consistent nodal force vector -- 8.4 Constant strain triangle (CST) element -- 8.5 Element defects -- 8.5.1 Constant strain triangle element -- 8.5.2 Bilinear rectangle (Q4) -- 8.6 Higher order elements -- 8.6.1 Quadratic triangle (linear strain triangle) -- 8.6.2 Q8 quadratic rectangle -- 8.6.3 Q9 quadratic rectangle -- 8.6.4 Q6 quadratic rectangle -- 8.7 Assembly, boundary conditions, solution, and postprocessing -- References -- Chapter 9 Finite element analysis of one-dimensional heat transfer problems -- 9.1 Introduction -- 9.2 One-dimensional heat transfer -- 9.2.1 Boundary conditions for one-dimensional heat transfer -- 9.3 Finite element formulation of the one-dimensional, steady state, heat transfer problem -- 9.3.1 Element equilibrium equations for a generic one-dimensional element -- 9.3.2 Finite element form with linear interpolation -- 9.4 Element equilibrium equations: general ordinary differential equation -- 9.5 Element assembly -- 9.6 Boundary conditions -- 9.6.1 Natural boundary conditions -- 9.6.2 Essential boundary conditions -- 9.7 Computer implementation -- Problems -- Chapter 10 Heat transfer problems in two-dimensions -- 10.1 Introduction -- 10.2 Solution domain and its boundary -- 10.3 The heat equation and its boundary conditions -- 10.3.1 Boundary conditions for heat transfer in two-dimensional domain -- 10.4 The weak form of heat transfer equation in two dimensions -- 10.5 The finite element form of the two-dimensional heat transfer problem -- 10.5.1 Finite element form with linear, quadrilateral (Q4) element -- 10.6 Natural boundary conditions -- 10.6.1 Internal edges -- 10.6.2 External edges subjected to prescribed heat flux -- 10.6.3 External edges subjected to convection -- 10.6.4 External edges subjected to radiation. |
| Subject |
Finite element method.
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Méthode des éléments finis.
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Finite element method
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| Other Form: |
Print version: 012821127X 9780128211274 (OCoLC)1240307316 |
| ISBN |
9780128232002 (electronic bk.) |
|
0128232005 (electronic bk.) |
|
9780128211274 |
|
012821127X |
| Standard No. |
AU@ 000072326360 |
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